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Updated Oct 7

Problem 938

Statement. Let $A=\{n_1<n_2<\cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$ ).

Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$ ?

Status. Open.

Source. erdosproblems.com/938, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #938, https://www.erdosproblems.com/938.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (12)

Diophantine Problems and Powers diophantine_problems/doorn_2026_three_term_arithmetic_progressions_consecutive_powerful Conjecture 5 (p. 6): (C_7 + o(1)) log n consecutive Pellian triples up to n Conjecture 7 (p. 8): each class A_i has order log n elements up to n Corollary 3 (p. 5): a sufficient condition for consecutive triples Lemma 2 (p. 4): counting powerful numbers between (x-2)^2 and x^2 Lemma 6 (p. 7): consecutive powerful progressions with two squares Theorem 1 (p. 2): powerful progressions with d = 2√N + 1 Theorem 4 (p. 5): inequality (6) infinitely often for one large modulus diophantine_problems/narumi_2025_number_k_full_integers_between_three Corollary 1: infinitely many triples of successive k-th powers that are consecutive k-full integers Theorem 1: density of n with prescribed k-full integers in two successive inter-power intervals

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